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SMALL - GAIN THEOREMS of
LASALLE TYPE for
HYBRID SYSTEMS
Daniel Liberzon (Urbana-Champaign)
Dragan Nešić (Melbourne)
Andy Teel (Santa Barbara)
CDC, Maui, Dec 2012
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MODELS of HYBRID SYSTEMS
…
[Goebel-Sanfelice-Teel]
Flow:
Jumps:
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HYBRID SYSTEMS as FEEDBACK CONNECTIONS
continuous
discrete
Every hybrid system can be thought of in this way
But this special decomposition is not always the best one
E.g., NCS:
network protocol
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HYBRID SYSTEMS as FEEDBACK CONNECTIONS
HS1
HS2
Can also consider external signals
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SMALL – GAIN THEOREM
Small-gain theorem [Jiang-Teel-Praly ’94] gives GAS if:
• (small-gain condition)
• Input-to-state stability (ISS) from to [Sontag ’89]:
• ISS from to :
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SUFFICIENT CONDITIONS for ISS
This gives “strong” ISS property [Cai-Teel ’09]
• For :
• For :
where
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LYAPUNOV – BASED SMALL – GAIN THEOREM
on•
(small-gain condition)•
is a Lyapunov function for the overall hybrid system
Then
Pick s.t.
on•
Assume:
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LYAPUNOV – BASED SMALL – GAIN THEOREM
Generalizes Lyapunov small-gain constructions for continuous [Jiang-Mareels-Wang ’96] and discrete [Laila-Nešić ’02] systems
decreases along solutions of the hybrid systemOn the boundary, use Clarke derivative
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LIMITATION
on•
on•
The strict decrease conditions
are often not satisfied off-the-shelf
E.g.:
Since and we would typically have
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LASALLE THEOREM
• all nonzero solutions have both flow and jumps
•
Assume:
•
•
As before, pick and let
Then is non-increasing along both flow and jumps
and it’s not constant along any nonzero traj. GAS
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SKETCH of PROOF
is nonstrictly decreasing along trajectories
Trajectories along which is constant? None!
GAS follows by LaSalle principle for hybrid systems [Lygeros et al. ’03, Sanfelice-Goebel-Teel ‘05]
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QUANTIZED STATE FEEDBACK
QUANTIZER
CONTROLLER
PLANT
Hybrid quantized control: is discrete state
– zooming variable
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QUANTIZED STATE FEEDBACK
QUANTIZER
CONTROLLER
PLANT
Hybrid quantized control: is discrete state
Zoom out to overcome saturation
– zooming variable
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QUANTIZED STATE FEEDBACK
QUANTIZER
CONTROLLER
PLANT
Hybrid quantized control: is discrete state
After the ultimate bound is achieved,recompute partition for smaller region
Can recover global asymptotic stability
– zooming variable
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SMALL – GAIN ANALYSIS
quantization error
Zoom in:
where
ISS from to with gain small-gaincondition!
ISS from to with some linear gain
Can use quadratic Lyapunov functions to compute the gains
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CONCLUSIONS
• Basic idea: small-gain analysis tools are naturally
applicable to hybrid systems
• Main technical results: (weak) Lyapunov function
constructions for hybrid system interconnections
• Applications:
• Quantized feedback control
• Networked control systems
• Event-triggered control [Tabuada]
• Other ???
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